Klaus Altmann
We investigate how the chain property for the associated primes of monomial degenerations of toric (or lattice) ideals can be generalized to arbitrary A-graded ideals. The generalization works in dimension d=2, but it fails for d>2.
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The asymptotic stability of several homological invariants of the graded pieces of a graded module has attracted quite a lot of attention over the last decades. We provide in this text several stability results together with estimates of the degree from which it stabilizes. Before we establish these regularity results, we prove several facts about depth and cohomological dimension with respect to a finitely generated ideal and about Castelnuovo-Mumford regularity of a graded module.
We establish a generalization of the Briancon-Skoda theorem about integral closures of ideals for graded systems of ideals satisfying a certain geometric condition.
We show the existence (and define) the mixed multiplicities of arbitrary graded families of ideals under mild assumptions. In particular, our methods and results are valid for the case of arbitrary $\mathfrak{m}$-primary graded families. Furthermore, we provide a far-reaching "Volume = Multiplicity formula" for the mixed multiplicities of graded families of ideals.
In this expository paper we survey results that relate Hilbert coefficients of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems of S. Huckaba and T. Marley. These were proved using homological techniques. We provide simple proofs using superficial sequences.
We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra.
Density functions for filtrations of graded ideals
本文发展了标准分次整环上任意分次理想滤链的密度函数一般理论，证明其基本性质、分段多项式逼近与饱和密度函数的可微性，并给出符号幂与整闭包重合的数值判据及混合重数不等式。
We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important cases, such as monomial ideals and toric rings. We prove that in these cases the Betti category is a finite combinatorial object that completely determines the structure of the minimal free resolution. For monomial ideals, the Betti category is the same as the Betti poset that we studied in a previous article. We describe in detail and with examples how the theo…
Given a graded sequence of ideals (a_m) on a smooth variety $X$ having finite log canonical threshold, suppose that for every m we have a divisor E_m over X that computes the log canonical threshold of a_m, and such that the log discrepancies of the divisors E_m are bounded. We show that in this case the set of divisors E_m is finite.
Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $γ> 0$ such that for all $n \ge 0$, $$\ell_R(R/I_{n+1}) - \ell_R(R/I_n) < γn^{d-1}.$$
We give simple necessary and sufficient conditions on projective schemes over a field k for asymptotic limits of the growth of all graded linear series of a fixed Kodaira-Iitaka dimension to exist. We also give necessary and sufficient conditions on a local ring R of dimension d for the limit \lim_{i\rightarrow\infty}\frac{\ell_R(R/I_n)}{n^d} to exist whenever \{I_n\} is a graded family of m_R-primary ideals. We give a number of applications.
Graded families of ideals and convex regions
研究\(\mathbb{K}\)-域中理想分次族与相关凸区域（牛顿 - 奥昆科夫区域）的相互作用，结合阿图什 - 韦茨拓扑技术和凸几何性质，通过区域包含关系刻画一对理想分次族的渐近复苏数。
Multi-graded Macaulay Dual Spaces
本文提出计算多分次理想Macaulay对偶空间的算法，通过自然对偶性将齐次理想运算推广到多分次情形，并利用全序递归构造基，附示例验证。
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ with maximal ideal $\frak{m}=(x_1,...,x_n)$, and let $I$ be a graded ideal of $S$. In this paper, we define the saturation number $\sat(I)$ of $I$ to be the smallest non-negative integer $k$ such that $I:\mm^{k+1}= I:\mm^k$. We show that $f(k)$ is linearly bounded, and that $f(k)$ is a quasi-linear function for $k\gg 0$, if $I$ is a monomial ideal. Furthermore, we show that $\sat(I^k)=k$ if $I$ is a principal Borel ideal and prove that $\sat(I_{d,n}^k) =\max\{l\:\; (kd-l)/(k-l) \leq n\},$ where $I_{d,n}$ is the …
We generalize a result of Ein-Lazarsfeld-Smith (math.AG/0202303), proving that for an arbitrary sequence of zero-dimensional ideals, the multiplicity of the sequence is equal with its volume. This is done using a deformation to monomial ideals. As a consequence of our result, we obtain a formula which computes the multiplicity of an ideal I in terms of the multiplicities of the initial monomial ideals of the powers I^m. We use this to give a new proof of the inequality between multiplicity and the log canonical threshold due to de Fernex, Ein and the author.
The graded Betti numbers are not tropical invariants
该研究证明分次Betti数不是齐次理想或赋值域上射影代数概形热带化的不变量，通过构造一组对应同一热带理想但Betti表不同的饱和理想完成论证。
We study algebras k[x_1,...,x_n]/I which admit a grading by a subsemigroup of N^d such that every graded component is a one-dimensional k-vector space. V.I.~Arnold and coworkers proved that for d = 1 and n <= 3 there are only finitely many isomorphism types of such A-graded algebras, and in these cases I is an initial ideal (in the sense of Groebner bases) of a toric ideal. In this paper it is shown that Arnold's finiteness theorem does not extend to n = 4. Geometric conditions are given for I to be an initial ideal of a toric ideal. The varieties defined by A-graded algebras are characterized…
Let R be a local Cohen-Macaulay ring, let I be an R-ideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G is not necessarily Cohen-Macaulay. We assume that I is either equimultiple, or has analytic deviation one, but we do not have any restriction on the reduction number. We also give a general estimate for the depth of G involving the first r+l powers of I, where r denotes the Castelnuovo regularity of G and l denotes the analytic spread of I.
Topological and scheme-theoretic properties of the $D$-graded Proj construction
本文研究了多分层Proj构造的拓扑和模论性质，提出D-素理想概念，并建立多分层的诺特定理，探讨了因子分层环的分离性条件及Serre扭的自由性判据。
We find simple necessary and sufficient conditions on a local ring $R$ of dimension $d$ for the limit $$ \lim_{i\rightarrow\infty}\frac{\ell_R(R/I_n)}{n^d} $$ to exist whenever $\{I_n\}$ is a graded family of $m_R$-primary ideals, and give a number of applications. We also give simple necessary and sufficient conditions on projective schemes over a field $k$ for asymptotic limits of the growth of all graded linear series of a fixed Kodaira-Iitaka dimension to exist.
We use \ZZ^d-gradings to study d-dimensional monomial ideals. The Koszul functor is employed to interpret the quasidegrees of local cohomology in terms of the geometry of distractions and to explicitly compute the multiplicities of exponents. These multigraded techniques originate from the study of hypergeometric systems of differential equations.
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over the field $K$, and let $I\subset S$ be a graded ideal. It is shown that for $k \gg0$ the postulation number of $I^k$ is bounded by a linear function of $k$, and it is a linear function of $k$, if $I$ is generated in a single degree. By using the relationship of the $h$-vector with the higher iterated Hilbert coefficients of $I^k$ it is shown that the Hilbert coefficients $e_i(I^k)$ of $I^k$ are polynomials for $k \gg 0$, whenever $I$ is generated in a single degree.
Resurgence number of graded families of ideals
研究格雷德理想家族的 resurgence 和渐近 resurgence 数，探讨其有限性和有理性的条件，以及如何通过 Rees 估值或实际极限计算，并分析积分闭包对渐近 resurgence 的影响。
Let X be a smooth projective variety with torsion-free Picard group. We introduce complexes of vector spaces whose homology determines the structure of the minimal free resolution of the Cox ring of X over the polynomial ring and show how the homology of these complexes can be studied by purely geometric methods. As an application of these techniques we give a simple new proof of a characterization of the Cox rings of Del Pezzo surfaces (of degree >1) conjectured by Batyrev and Popov.
We find formulas for the graded core of certain m-primary ideals in a graded ring. In particular, if S is the section ring of an ample line bundle on a Cohen-Macaulay complex projective variety, we show that under suitable hypothesis, the core and graded core of the ideal of S generated by all elements of degrees at least N (for some, equivalently every, large N) are equal if and only if the line bundle admits a non-zero global section. We also prove a formula for the graded core of the powers of the unique homogeneous maximal ideal in a standard graded Cohen-Macaulay ring of arbitrary charact…
Homogeneity of zero-divisors, units and colon ideals in a graded ring
本文将Kaplansky零因子猜想和单位猜想推广至G-分次环，证明了全序群分次环中零因子、可逆元素及冒号理想的齐次性，并推广了McCoy和Armendariz的经典定理。
In this paper we show that the sets of $F$-jumping coefficients of ideals form discrete sets in certain graded $F$-finite rings. We do so by giving a criterion based on linear bounds for the growth of the Castelnuovo-Mumford regularity of certain ideals. We further show that these linear bounds exists for one-dimensional rings and for ideals of (most) two-dimensional domains. We conclude by applying our technique to prove that all sets of $F$-jumping coefficients of all ideals in the determinantal ring given as the quotient by $2\times 2$ minors in a $2\times 3$ matrix of indeterminates form d…
Given $Σ\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |Σ|$, we prove that $I_a(Σ)$, the ideal generated by all $a$-fold products of $Σ$, has linear graded free resolution. This allows us to determine a generating set for the defining ideal of the Orlik-Terao algebra of the second order of a line arrangement in $\mathbb P_{\mathbb{K}}^2$, and to conclude that for the case $k=3$, and $Σ$ defining such a line arrangement, the ideal $I_{|Σ|-2}(Σ)$ is of fiber type. We …
This paper introduces two new notions of graded linear resolution and graded linear quotients, which generalize the concepts of linear resolution property and linear quotient for modules over the polynomial ring $A=k[x_1, \dots ,x_n]$. Besides, we compare graded linearity with componentwise linearity in general. For modules minimally generated by a regular sequence in a maximal ideal of $A$, we show that graded linear quotients imply graded linear resolution property for the colon ideals. On the other hand, we provide specific characterizations of graded linear resolution property for the Stan…
Multiplicities of graded families of ideals on Noetherian local rings
本文研究了Noetherian局部环上分级理想族的乘数概念，推广了传统理想乘数，并证明了多个经典定理在该框架下的泛化，包括混合乘数、Rees定理和Minkowski不等式。
The $i$-th local cohomology module of a finitely generated graded module $M$ over a standard positively graded commutative Noetherian ring $R$, with respect to the irrelevant ideal $R_+$, is itself graded; all its graded components are finitely generated modules over $R_0$, the component of $R$ of degree 0. This paper is concerned with the asymptotic behaviour of $\Ass_{R_0}(H^i_{R_+}(M)_n)$ as $n \to -\infty$. The smallest $i$ for which such study is interesting is the finiteness dimension $f$ of $M$ relative to $R_+$, defined as the least integer $j$ for which $H^j_{R_+}(M)$ is not finitely …
The Proj of the Rees algebra of a graded family of ideals
研究Rees代数的渐进代数的Proj在Noetherian局部环中为Noetherian的条件，探讨高维情形下该性质不成立的例子及纤维锥的极小素理想有限性。
Let $G$ be a finitely generated abelian group, and let $S = A[x_1, ..., x_n]$ be a $G$-graded polynomial ring over a commutative ring $A$. Let $I_1, ..., I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded $S$-module. We show that, when $A$ is Noetherian, the nonzero $G$-graded Betti numbers of $MI_1^{t_1} ... I_s^{t_s}$ exhibit an asymptotic linear behavior as the $t_i$s get large.
Let $R=k[x_1, ..., x_n]$ be a polynomial ring and let $I\subset R$ be a graded ideal. In \cite{R}, Römer asked whether under the Cohen-Macaulay assumption the $i$-th Betti number $β_{i}(R/I)$ can be bounded above by a function of the maximal shifts in the minimal graded free $R$-resolution of $R/I$ as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen-Macaulay algebras $k[x_1, ..., x_n]/I$ when $I$ is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds ar…
A central question in liaison theory asks whether every Cohen-Macaulay, graded ideal of a standard graded K-algebra belongs to the same G-liaison class of a complete intersection. In this paper we answer this question positively for toric ideals defining edge subrings of bipartite graphs.
In this article we extend a previous definition of Castelnuovo-Mumford regularity for modules over an algebra graded by a finitely generated abelian group. Our notion of regularity is based on Maclagan and Smith's definition, and is extended first by working over any commutative base ring, and second by considering local cohomology with support in an arbitrary finitely generated graded ideal $B$, obtaining, for each $B$, a $B$-regularity region. The first extension provides a natural approach for working with families of sheaves or of graded modules, while the second opens new applications. We…
We study the annihilator of the cokernel of a map of free Z/2-graded modules over a Z/2-graded skew-commutative algebra in characteristic 0 and define analogues of its Fitting ideals. We show that in the ``generic'' case the annihilator is given by a Fitting ideal, and explain relations between the Fitting ideal and the annihilator that hold in general. Our results generalize the classical Fitting Lemma, and extend the key result of Green [1999]. They depend on the Berele-Regev theory of representations of general linear Lie super-algebras.
We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives an algebraic formulation of the minimum distance of a projective Reed-Muller-type code in terms of the algebraic invariants and structure of the underlying vanishing ideal. Then we give a method, based on Groebner bases and Hilbert functions, to find lower bounds for the minimum distance of certain Reed-Muller-type codes. Finally we show explicit upper bound…
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